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First-Order Logic with Isomorphism FIRST-ORDER LOGIC WITH ISOMORPHISM DIMITRIS TSEMENTZIS Abstract. The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality (FOL=) allows us to define structures on sets. We develop the syntax, semantics and de- ductive system for such a logic, which we call first-order logic with isomorphism (FOL=∼). The syntax of FOL=∼ extends FOL= in two ways. First, by incorporating into its signatures a notion of dependent sorts along the lines of Makkai’s FOLDS as well as a notion of an h-level of each sort. Second, by specifying three new logical sorts within these signatures: isomorphism sorts, reflexivity predicates and trans- port structure. The semantics for FOL=∼ is then defined in homotopy type theory with the isomorphism sorts interpreted as identity types, reflexivity predicates as relations picking out the trivial path, and transport structure as transport along a path. We then define a deductive system D=∼ for FOL∼= that encodes the sense in which the inhabitants of isomorphism sorts really do behave like isomorphisms and prove soundness of the rules of D=∼ with respect to its homotopy semantics. Fi- nally, as an application, we prove that precategories, strict categories and univalent categories are axiomatizable in FOL∼=. Contents Introduction 1 1. Preliminaries on FOLDS 3 2. Semantics of FOLDS in MLTT 14 3. Syntax of FOL=∼ 25 4. Homotopy Semantics of FOL=∼ 35 5. Deductive System for FOL=∼ 47 6. Examples and Applications 55 References 61 arXiv:1603.03092v2 [math.LO] 25 Sep 2017 Introduction The Univalent Foundations of Mathematics (UF) [Uni13] take their basic objects to be homotopy types. In UF mathematical structures are therefore encoded as struc- tured homotopy types, similar to how in set-theoretic foundations they are encoded as structured sets. This basic picture allows us to envision a model theory in which Date: September 27, 2017. 2010 Mathematics Subject Classification. 03G99, 03B15, 03B22, 03C99. Key words and phrases. Univalent Foundations, Categorical Logic, Homotopy Type Theory. 1 2 DIMITRIS TSEMENTZIS mathematical objects are no longer described as structured sets, but rather as struc- tured homotopy types. In this paper we develop a logic through which this process can be carried out. We will call it first-order logic with isomorphism (FOL=∼) in order to reflect the following fundamental idea of UF: the primitive notion of equality in UF is itself a structure that behaves like the structure of isomorphisms between structures, rather than the fact of identity between sets. The syntax of FOL=∼ will be based on an extension of the syntax of Makkai’s FOLDS (First-Order Logic with Dependent Sorts) [Mak95]. The signatures of FOLDS can be understood as inverse categories where the arrows encode variable dependencies between the objects (understood as “sorts”). The key addition of FOL=∼ to FOLDS is to add a notion of an “h-level” to each sort as well as notions of “isomorphism sorts”, “reflexivity predicates” and “transport structure” understood as logical sorts (i.e. with a fixed denotation) defined on top of the non-logical sorts in accordance with the latter’s h-level. It is helpful to think of the syntax of FOL=∼ as standing to FOLDS in the same relation that the syntax of first-order logic with equality (FOL=) stands to plain first-order logic. To add equality to a first-order signature Σ, one simply adds a binary relation with a certain fixed denotation. The analogous process for a FOLDS signature L is carried out in terms of an “isomorphism completion” operation on inverse categories which attaches logical sorts to pre-existing sorts in L in a manner compatible with their h-level. The intended semantics for FOL=∼ is in UF, where in particular “isomorphism sorts” are interpreted as path spaces (if one thinks in terms of abstract homotopy theory) or identity types (if one thinks in terms of homotopy type theory). The basic idea of the “homotopy semantics” of FOL=∼ is the following: non-logical sorts of h-level m are interpreted as (dependent functions landing in) types of h-level m and we make this precise by interpreting FOL=∼ into a version of “book HoTT”, i.e. the formal system outlined in [Uni13]. The isomorphism sorts that have been added to the syntax of FOL=∼ are then interpreted as identity types, the reflexivity predicates are interpreted as predicates picking out refl and transport structure is interpreted as (a relational version of) the transport function trans induced by a given path. Finally, we present a deductive system D=∼ on the syntax of FOL=∼ that can justify its homotopy semantics. It is based on a standard sequent calculus for first-order logic to which we add three rules: a rule asserting the existence of the “reflexivity” isomorphisms, a rule specifying that “transport along reflexivity” does nothing and, crucially, a propositional version of the J-rule of MLTT. We can then prove that the deductive system is sound with respect to the above-sketched semantics, which is the main result of this paper. Related Work. Our work connects to several strands of ongoing work in UF. The connection between inverse categories and homotopy theory is well-known and has been developed extensively by Shulman in [Shu15a,Shu15b] and our work builds on this connection, albeit from a more syntactic viewpoint, rather than within the categorical semantics of HoTT. Our framework also provides a general definition of a FIRST-ORDERLOGICWITHISOMORPHISM 3 (logical) signature for UF and could thus be used to generalize the Structure Identity Principle of [Uni13] as indeed has been sketched in [ANS14, Tse17]. Formalizations of category theory in the style of FOLDS has also been carried out by Ahrens in UniMath [VAG+] and his formalization overlaps with some of the material of Sec- tion 6. Furthermore, our system FOL=∼ (as well as FOLDS) can be thought of as a dependently-typed first-order logic with specific features that make it suitable for UF, and in that regard it connects to the work of Palmgren [Pal16]. Finally, although we do not (yet) adopt this point of view, FOL=∼ can be thought of as a proposal to add a “logic layer” to HoTT along the lines of so-called logic-enriched type theory [GA06] and therefore is connected to the work of [PL15]. Outline of the Paper. In Section 1 we introduce the syntax of FOLDS as well as a proof system DFOLDS for this syntax and state some preliminary results. In Section 2 we define an interpretation of the syntax of FOLDS into MLTT via a direct interpretation of the raw syntax of the former system into the raw syntax of the latter. In Section 3 we build on the syntax of FOLDS to define the syntax of FOL=∼. In Section 4 we define the homotopy semantics of FOL=∼ by defining an interpretation of the syntax of FOL=∼ into HoTT, building on the interpretation of FOLDS into MLTT. In Section 5 we define a proof system D=∼ for FOL=∼ and prove soundness for D=∼ with respect to our homotopy semantics. Finally, in Section 6 we show how precategories, strict categories and univalent categories can be axiomatized in terms of FOL=∼. 1. Preliminaries on FOLDS We will assume familiarity with the basics of categorical logic as well as of in- tensional Martin-Löf Type Theory and its homotopy interpretation. For category- theoretic background [ML98] remains the standard reference; for the basics of de- pendent type theory we recommend [ML84, Hof97]; for the homotopy interpretation and an introduction to the Univalent Foundations see [Uni13,KLV14] and references therein. We will now present in more detail the basic syntax of FOLDS inspired by Makkai’s original presentation in [Mak95]. Definition 1.1 (Inverse Category). An inverse category is a category L such that (1) L has no non-identity endomorphisms (2) L is skeletal (3) For any object K there is a finite number of arrows with domain K Remark 1.2. An inverse category is so-named because it can be thought of as a category in which all arrows “go in one direction” and in particular a category in which all arrows “point downwards”. Condition (3) in Definition 1.1 (which Makkai called “finite fan-out”) ensures that every object in an inverse category is only a “finite path” away from an object at the “bottom”. This allows us to carry out inductive definitions on inverse categories as indeed we will be doing throughout this paper. Notation. We write N∞ for the set N ∪ {∞}. We will sometimes write K ∈ L as an abbreviation for K ∈ ObL and we write L(K,K′) for the morphisms from K to 4 DIMITRIS TSEMENTZIS ′ ′ K in L. For any arrow f : K → K in an inverse category L we will write Kf for its codomain K′. For any object K ∈ ObL we write K//L for the set (indeed, cosieve) of non-identity arrows with domain K. The objects of any inverse category L can be stratified into levels, defined (inductively) as follows: 0 if K is the domain only of 1 K l(K)= sup l(Kf ) + 1 otherwise f∈K//L Terminology. We will call l(K) the level of K. We will generally refer to the objects of L as sorts. We call a non-identity arrow f : K → A top-level (for K) if it does not factor through another arrow, i.e. if there is no non-identity arrows h, k ∈ MorL such that f = hk and we write top(K) for the set of top-level arrows of a sort K.
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